Fluid Mechanics

JEE Main level, numerical answer, medium

122 of 192

Question 122 of 192

A spherical drop of liquid splits into 10001000 identical spherical drops. If ui{u}_{i} is the surface energy of the original drop and uf{u}_{f} is the total surface energy of the resulting drops, the (ignoring evaporation), ufui=(10x)\dfrac{{u}_{f}}{{u}_{i}}=\left(\dfrac{10}{x}\right). Then value of xx is ______:

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Answer: 1

Let RR be the radius of bigger drop and rr be the radius of smaller drop.
Here, volume will remain conserved, so VR=Vr{V}_{R}={V}_{r}.
⇒43πR3=1000×43πr3\Rightarrow \dfrac{4}{3}\pi {R}^{3}=1000\times \dfrac{4}{3}\pi {r}^{3}
⇒R=10r\Rightarrow R=10r
As surface energy is stated as u=SAu=SA, here, SS is surface tension and AA is surface area.
Thus, ui=S(4πR2){u}_{i}=S\left(4\pi {R}^{2}\right) and uf=S(4πr2)×1000{u}_{f}=S\left(4\pi {r}^{2}\right)\times 1000
The ratio is ufui=1000r2R2\dfrac{{u}_{f}}{{u}_{i}}=\dfrac{1000{r}^{2}}{{R}^{2}}
⇒ufui=101\Rightarrow \dfrac{{u}_{f}}{{u}_{i}}=\dfrac{10}{1}
Hence, the value of x=1x=1.

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